The magnetohydrodynamic (MHD) Couette flow of two immiscible fluids in a horizontal channel with isothermal walls in the presence of an applied electric and inclined magnetic field has been investigated in the paper. Both fluids are electrically conducting, while the channel plates are electrically insulated. The general equations that describe the discussed problem under the adopted assumptions are reduced to ordinary differential equations, and closed-form solutions are obtained in both fluid regions of the channel. Separate solutions with appropriate boundary conditions for each fluid have been obtained, and these solutions have been matched at the interface using suitable matching conditions. The analytical results for various values of the Hartmann number, the angle of magnetic field inclination, loading parameter, and the ratio of fluid heights have been presented graphically to show their effect on the flow and heat transfer characteristics.

The flow and heat transfer of electrically conducting fluids in channels and circular pipes under the effect of a transverse magnetic field occurs in magnetohydrodynamic (MHD) generators, pumps, accelerators, and flowmeters and have applications in nuclear reactors, filtration, geothermal systems, and others

The interest in the outer magnetic field effect on heat-physical processes appeared seventy years ago. Blum et al. [

Yang and Yu [

All the mentioned studies pertain to a single-fluid model. Most of the problems relating to the petroleum industry, geophysics, plasma physics, magneto-fluid dynamics, and so forth involve multifluid flow situations. Hartmann flow of a conducting fluid and a non-conducting fluid layer contained in a channel has been studied by Shail [

There have been some experimental and analytical studies on hydrodynamic aspects of the two-fluid flow reported in the recent literature. Following the ideas of Alireza and Sahai [

Recent studies show that magnetohydrodynamic (MHD) flows can also be a viable option for transporting weakly conducting fluids in microscale systems, such as flow inside the microchannel networks of a lab-on-a-chip device [

MHD flows inside channels can be propelled in many different ways, for example, in electromagnetohydrodynamics (EMHDs) axial flow along a channel is generated by the interaction between the magnetic field and an electric field acting normal to it. Regardless of the purpose of a multifluid EMHD flow, it is important to understand the dynamics of interfaces between the fluids and its effect on the transport characteristics of the system. Keeping in view the wide area of practical importance of multifluid flows as mentioned above, it is the objective of the present study to investigate the MHD Couette flow and heat transfer of two immiscible fluids in a parallel-plate channel in the presence of applied electric and inclined magnetic fields.

As mentioned in the introduction, the problem of the EMHD Couette two fluid flow has been considered in this paper. The fluids in the two regions have been assumed immiscible and incompressible, and the flow has been steady, one-dimensional, and fully developed. Furthermore, the two fluids have different kinematic viscosities

Physical model and coordinate system.

A uniform magnetic field of the strength

The fluid velocity, treating the problem as a monodimensional, and the magnetic field distributions for the case of inclined and induced magnetic field [

Finally the continuity, momentum, and induction equation written in the classic quasi-static low magnetic Reynolds number approximation [

The fluid and thermal boundary conditions have been unchanged by the addition of electromagnetic fields. The no-slip conditions require that the fluid velocities are equal to the plate’s velocities, and boundary conditions on temperature are isothermal conditions. In addition, the fluid velocity, sheer stress, induced magnetic field, induced magnetic flux (induced currents at the interface of conductors [

The governing equation for the velocity

Once the velocity distributions were known, the temperature distributions for the two regions have been determined by solving the energy equation subject to the appropriate boundary and interface conditions (

Recent technological trends show that the use of external fields to generate the flow inside channels, such as electrohydrodynamic, MHD, and electrokinetic flows, can be more advantageous in many microscale applications. In order to show the results of the considered MHD Couette flow problem graphically, two fluids important for technical practice (selected for the development of MHD pump under the project TR35016) have been chosen, and the parameters

Velocity profiles for different values of inclination angle

Figures

Figure

Temperature profiles for different values of inclination angle

Ratio of an induced and externally imposed magnetic field

It can be seen from Figures

Figure

Figures

Velocity profiles for different values of Hartmann numbers

The effect of increasing the Hartmann number on temperature profiles (Figure

Temperature profiles for different values of Hartmann numbers

Ratio of an induced and externally imposed magnetic field for different values of Hartmann numbers

The influence of the Hartmann number had quite similar effect on the ratio of induced and externally applied magnetic field as shown in Figure

The influence of the induced magnetic field in the considered case is not so important, but in similar flow problems where the transversal component velocity is present, the knowledge of the imposed and induced field ratio can have great significance.

Of particular significance is the analysis when the loading factor

Temperature profiles for different values of loading factor

Figure

Velocity profiles for different values of loading factor

The obtained results show that different values of the inclination angle, the Hartmann number, and the loading factor are a convenient control method for heat and mass transfer processes.

The ratio of an induced and externally imposed magnetic field had a considerable change when the loading parameter was different from zero, especially in region II.

Figure

Ratio of an induced and externally imposed magnetic field for different values of loading factor

The effect of the ratio of heights of the two regions on the velocity field is shown in Figure

Velocity profiles for different values of height ratio

The effect of ratio of the heights of the two regions on temperature field is same as its effect on velocity field, which is evident from Figure

Temperature profiles for different values of height ratio

Ratio of an induced and externally imposed magnetic field for different values of height ratio

Temperature profiles for different values of Eckert number

Figure

The problem of MHD Couette flow and heat transfer of two immiscible fluids in a horizontal parallel-plate channel in the presence of applied electric and inclined magnetic fields was investigated analytically. Both fluids were assumed to be Newtonian, electrically conducting, and have constant physical properties. Separate closed form solutions for velocity, temperature, and magnetic induction of each fluid were obtained taking into consideration suitable interface matching conditions and boundary conditions. The results were numerically evaluated and presented graphically for two fluids important for technical practice. Only part of the results are presented for various values of the magnetic field inclination angle, Hartmann number, loading parameter, and ratio of fluid heights in region I and II.

Furthermore, it was concluded that the flow and heat transfer aspects of two immiscible fluids in a horizontal channel with insulating walls can be controlled by considering different fluids having different viscosities and conductivities and also by varying the heights of regions. The obtained results show also that different values of the inclination angle, the Hartmann number, and the loading factor are a convenient control method for heat and mass transfer processes.

This paper is supported by the Serbian Ministry of Sciences and Technological development (Project no. TR 35016; Research of MHD flow in the channels, around the bodies and application in the development of the MHD pump). The authors wish to thank the reviewer for his careful, unbiased, and constructive suggestions that significantly improved the quality of this paper.